## Algebraic & Geometric Topology

### The Ptolemy field of $3$–manifold representations

#### Abstract

The Ptolemy coordinates for boundary-unipotent $SL(n, ℂ)$–representations of a $3$–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the $A$–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) $PSL(2, ℂ)$-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 371-397.

Dates
Revised: 9 May 2014
Accepted: 7 July 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510840915

Digital Object Identifier
doi:10.2140/agt.2015.15.371

Mathematical Reviews number (MathSciNet)
MR3325740

Zentralblatt MATH identifier
1322.57018

Subjects
Secondary: 57M27: Invariants of knots and 3-manifolds

#### Citation

Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. The Ptolemy field of $3$–manifold representations. Algebr. Geom. Topol. 15 (2015), no. 1, 371--397. doi:10.2140/agt.2015.15.371. https://projecteuclid.org/euclid.agt/1510840915

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